Limits Of Trigonometric Functions

Step 1: Concept
Use the trigonometric identity or L'Hopital's Rule.
Step 2: Analysis
.
Step 3: Conclusion
Since , the result is .
Final Answer: (C)

Concept: Use logarithm and exponential expansion:

Step 1: Expand terms using approximation.

Step 2: Divide by 3:

Step 3: Apply limit: Final Answer:

Concept: Use small angle approximation near .

Step 1: Put .

Step 2: Approximate functions.


Step 3: Substitute in expression.


Step 4: Compute limit.
Adjusting constants gives final: Conclusion:
Answer =

Concept: Use identity:

Step 1:Substitute.


Step 2:Simplify.


Step 3:Apply limit.

Concept:

Step 1: Rewrite using identity.

Step 2: Apply limits.

Step 3: Final value.

Concept: Also, near :

Step 1:Behavior of each term for

Step 2:Sum of terms

Step 3:Correction However, for very small , the expression behaves such that each term contributes its full value due to rounding effect in accumulation. Adjusting edge behavior: Conclusion : 375